IB Mathematics (SL)/Functions and Equations

Topic 2 - Functions and Equations

Introduction

The general aim of this section is to explore the notion of function as a unifying theme. Additionally, candidates should be able to apply functional methods to a variety of situations. Use of a GDC (Graphing Display Calculator) is expected.

Concept of a Function

Inverse Function

The inverse function, is as its name signifies, the inverse of a function, shown as f ^{− 1}(x). This is accomplished by substituting x and y for one another within the equation, and evaluating the function to where you aim to get the y variable alone, again.

Examples:

Ex.1

$f(x)=x^2\,\!$

$y=x^2\,\!$

$(x)=(y)^2\,\!$

$\sqrt{x}=y\,\!$

$f\,\!$ ^{− 1}$(x)=\sqrt{x}\,\!$

Ex.2

$f(x)=3x-2\,\!$

$y=3x-2\,\!$

$(x)=3(y)-2\,\!$

$x+2=3y\,\!$

$\frac{x+2}{3}=y\,\!$

$f\,\!$ ^{− 1}$(x)=\frac{x+2}{3}\,\!$

The Graph of a Function

Horizontal and Vertical Asymptotes

An asymptote can be described as a line that represents the end behavior of a function. While they may be crossed, they may not be crossed at an infinite number of points. They can be horizontal, vertical, or oblique (diagonal).

For instance, if you look at a visual representation of y = 1 / x you will see that while the graph approaches the x-axis, or the line y = 0 it will never touch the line.

Exponents of the Variables

When dealing with a function, it s always a good idea to take a look at the highest power the variable(s) are to, among other things. For example, for the equations of y = x and y = x² the behaviors of these functions differ greatly. With y = x the function extends from negative infinty from Quadrant III to positive infinity in Quadrant I. While with the function of y = x² the function extends from Quadrant II to Quadrant I.

The Reciprocal Function

X ----> 1/x, i.e. f(x) = 1/x is defined as the reciprocal function.

Notice that:

- f(x) = 1/x is meaningless when x = 0 - the graph of f(x) = 1/x exists in the first and third quadrants only - f(x) = 1/x is symmetric about y = x and y = -x - f(x) = 1/x is asymptotic (approaches) to the x-axis and to the y-axis

(Source: Mathematics for the international student, Mathematics SL, International Baccalaureate Diploma Programme by John Owen, Robert Haese, Sandra Haese, Mark Bruce)

The Quadratic Function

Standard Form

$y = ax^2 + bx + c\,\!$

Vertex or Turning Point Form

$y = a(x-h)^2 + k\,\!$ , where (h,k) is the vertex

$(h,k) = (\frac{-b}{2a},\frac{4ac-b^2}{4a})$

Axis of Symmetry

-b/(2a)

Roots of the equation

$\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Discriminant

Discriminant=b²-4ac

Exponential Function

In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable. The exponential function is also often written as exp(x), especially when x is an expression complicated enough to make typesetting it as an exponent unwieldy.

The graph of y = ex is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some older sources refer to the exponential function as the anti-logarithm.

Sometimes the term exponential function is used more generally for functions of the form cbx, where the base b is any positive real number, not necessarily e.